A generalization of Sylvester's law of inertia (Q5955664)

The authors introduce the class of unitoid matrices as those that are diagonalizable by congruence; the nondegenerate canonical angles of a unitoid matrix \(A\) are the directions of the nonzero entries of a diagonal matrix congruent to \(A\). Sylvester's law states that two Hermitian matrices of the same size are congruent if and only if they have the same numbers of positive (respectively, negative) eigenvalues. The following is the primary result:NEWLINENEWLINENEWLINETwo unitoid matrices are congruent if and only if they have the same nondegenerate canonical angles.NEWLINENEWLINENEWLINEThe paper ends with two remarks. The authors close by noting that the canonical angles of \(A\) are, generally, unrelated to the eigenvalues of the unitary part in the polar decomposition of \(A\). They also note that their theorem may be proven using the canonical pair form for two Hermitian matrices.